Suppose I asked you

‘State a rule that governs (determines/characterizes) how people do make decisions’…

I want this rule to be…

1. Informative (it rules out at least some sets of choices)

2. Predictive (people rarely if ever violate this rule)

Similar question:

‘State a rule that governs how people should make decisions’..

By ‘should’ I mean that they will not regret having made decisions in this way.

2 minutes: discuss with your neighbour

If people did follow these rules, what would this imply and predict?

Rules defined as ‘axioms about preferences’

‘Standard axioms’ \(\rightarrow\) (imply that) choices can be expressed by ‘individuals maximising utility functions subject to their budget constraints’

\(\rightarrow\) yields predictions for individual behavior, markets, etc.

How is this used?

Utility will be expressed as a single number that arises from the combination of all goods and services consumed.

Utility from two goods

\[Utility = U(X,Y; other)\]

\[Utility = U(X,Y)\]

Maths revision: \(U(X,Y)\) expresses a function with two arguments, X and Y.

\(U(X,Y)\) must take some value for every positive value of X and Y.

Measuring and comparing utilities

Utility is not ‘observable and measurable in utils’

• (Unlike midi-chlorians or thetans)

• Utility is seen to govern an individual’s choices and thus it’s only inferred indirectly, from the choices people make

Revealed preference: if Al buys a cat instead of a dog, and a dog was cheaper, we assume Al gets more utility from a cat

Interpersonal comparisons are difficult

• Who gets ‘more’ utility?

…Interpersonal comparisons are difficult

• Transfer from Al to Betty: Is the reduction in Al’s utility more or less than the increase in Betty’s?

1. Completeness

\[A \succ B, B \succ A, \: or \: A \sim B \]

Fancy notation: Either A preferred to B, B preferred to A, or A indifferent to B

2. Transitivity

\[ A \succ B \: and \: B \succ C \rightarrow A \succ C \]

• Similar for indifference (\(\sim\))

If I prefer an Apple to a Banana and a Banana to Cherry

3. More is better

If the product is a ‘bad’ (e.g., pollution), redefine as the absence of the product

Who cares?

If people obey the first two assumptions (axioms),\(^\ast\) they will make choices in a way consistent with maximising a (continuous) utility function

*(and also ‘continuity’, which you can ignore for this module)

• How can we compare the “?” areas? Which are preferred?

• \(\rightarrow\) Compare utilities, depict using Indifference Curves

Indifference curves

Indifference curve

A curve that shows all the combinations of goods or services that provide the same level of utility

Credit: www2.econ.iastate.edu

Credit: Frank’s Economics on the web (MIT)

Q1: How do we know \(E \succ B\) ?

Back to fig 2.2 (board or visualiser)

• A to B: willing to give up 2 hamburgers to get 1 more soda.

• \(\rightarrow\) slope \(-2\), \(MRS=2\)

• From B to C? (think about it)

• …willing to give up 1 hamburger to get 1 more soda \(\rightarrow MRS =1\)

• C to D?

• \(MRS = \frac{1}{2}\)

Preference for variety/balance

Indifference curve map

Indifference curves never cross! And are never upwards sloping!

Illustrating particular preferences (NS fig 2.5)

App 2.3: Product positioning in marketing (read at home, see handout)

Perfect substitutes

Perfect substitutes

Goods A and B are Perfect Substitutes when an individual’s utility is linear in these goods

Perfect complements

Goods A and B are Perfect Complements when an individual only gains utility from (more) A if she also consumes a defined (additional) amount of B, and vice-versa

Budget constraint algebra

If I spend all my income (I will do over a ‘relevant lifetime’):

Expenditure on X + Expenditure on Y = Income (I)

\[P_X X + P_Y Y = I \]

\[Y = -\frac{P_X}{P_Y} X + \frac{I}{P_Y}\]

If slope budget constraint = slope indifc curve at point X,Y \(\rightarrow\) \[P_X/P_Y = MRS(X,Y)\]

• Warning: This equality holds at an optimal choice; it doesn’t hold everywhere.

At an optimal consumption choice (given above assumptions)

• Consume all of income; locate on budget line; follows from ‘more is better’

• Psychic tradeoff (MRS) equals market tradeoff (\(P_X/P_Y\)) if consuming both goods

Psychic tradeoff (MRS) equals market tradeoff (\(P_X/P_Y\)) if consuming both goods

Key intuition:

• If I can give up X for (buy less X, get more Y) at some rate, and the benefit I get from doing this is at a different rate,

• …then I can make myself better off.

• Thus the original point could not have been optimal.

More insight (mathy: ignore if this freaks you out)

Recall \(U=U(X,Y)\).

\[U_X(X,Y) := MU_X(X,Y)\]

Derivative w/ respect to X: rate utility increases if we add a little X, holding Y constant

… Similarly for \(MU_Y\).

• MRS: ‘how much Y would I be willing to give up to get a unit of X’?

• Ans: Depends on marginal benefit of each … we can show \(MRS(X,Y)=\frac{MU_{X}}{MU_{Y}}\)

Rearranging the utility maximising condition yields more intuition:

\[P_X/P_Y = MRS = MU_X/MU_Y\]

(at each consumption point X,Y)

• \[\frac{MU_X}{P_X} = \frac{MU_Y}{P_Y}\]

• Same ‘bang for each buck’ (if optimising)

Caveat on ‘corner solutions’

• If you are consuming both goods and optimising, \(P_X/P_Y = MRS = MU_X/MU_Y\) must hold

Advanced: This is a ‘necessary but not sufficient condition’, sufficient if DMRS everywhere

But…

But…

• If you are consuming both goods and optimising, \(P_X/P_Y = MRS = MU_X/MU_Y\) must hold

• But you might consume none of some good (say X):

• if even with no X, \(MU_X/P_X<MU_Y/P_Y\), the marginal utility of the first unit ‘per pound’ is lower

App 2.5: What’s a rich uncle’s promise worth?

Move to ppt slides here beginning with ‘Utility Maximization: A Graphical View’

Consider: are these ‘perfect substitutes’ for someone who wants caffeine, but has no taste buds?

Perfect substitutes, but not identical, e.g.,

\[U(X,Y)=4X+3Y\]

Rates each increase utility per-unit (derivative) are constant: \(MU_X = 4\), \(MU_Y = 3\)

So (for perfect substitutes) buy the one that increases it more *per-

With perfect substitutes: ‘Bang for the buck’ rule

\[U(X,Y)=4X+3Y\]

• Compare \(MU_X/P_X\) to \(MU_Y/P_Y\)

• Here, if \(4/P_X > 3/P_Y\), then buy X

• if \(4/P_X < 3/P_Y\), then buy Y; (if equal, buy either)

• Rearranging, if \(P_X < 4/3 P_Y\), buy X … etc.

Warning: If not perfect substitutes, MU ratios depend on consumption levels.

Perfect complements

Mathematical function example:

\[U(X,Y)=min(2X,Y)\]

E.g., X: bicycle frames, Y: wheels.

Warning: this min function looks backwards, but it’s correct; see notes

• Shortcut: figure out the proportions it will be consumed in
  • determine cost of ‘1 bundle of the combo’ at given prices
  • … then buy as many such bundles as you can afford

Middle-ground (*)

A Cobb-Douglas example

\[ U(X,Y)=\sqrt(XY) \]

\[MU_X = \frac{\partial}{\partial X} (XY)^{1/2} = \frac{1}{2} (Y/X)^{1/2}\]

\[MU_Y = \frac{1}{2} (X/Y)^{1/2}\]

Here, amount of Y you’d give up to get a unit of X:

\[MRS(X,Y)= MU_X/MU_Y = Y/X\]

{Check reasonable: The more Y I’ve , the more Y I’d give up to get another X :)}

…Cobb-Douglas ctd

\[MRS(X,Y)= MU_X/MU_Y = Y/X\]

Here utility-maximization requires, at optimal choices of X and Y: \[MRS(X,Y)= Y/X = P_X/P_Y\]

For any price ratio, find ratio of Y & X.

With prices and income, \(I\), find consumption of X & Y.

Rearranging optimization condition:

\[Y P_Y = X P_X\]

Optimization condition for this particular utility function:

\[Y P_Y = X P_X\]

Combining this with the budget constraint \[P_X X + P_Y Y = I\]

solve for X & Y, as fncns of prices & income (see notes) \(\rightarrow\)

\[Y = I/(2P_Y)\] \[X = I/(2P_X)\]